DOCSLIB.ORG

Lab 4 Fourier Series and the Gibbs Phenomenon

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lab 4 Fourier Series and the Gibbs Phenomenon Lab 4 Fourier Series and the Gibbs Phenomenon EE 235: Continuous-Time Linear Systems Department of Electrical Engineering University of Washington This work1 was written by Amittai Axelrod, Jayson Bowen, and Maya Gupta, and is licensed under the Creative Commons Attribution License.2 1 Introduction In this lab you will examine the Fourier series representation of periodic signals using mat- lab. In particular, you’ll study the truncated Fourier series reconstruction of a periodic function. 2 Useful matlab Commands In addition to what you already know about matlab, you may need to use the following commands. You’ve seen them in Lab 1, but use help to remind yourself of their syntax. • abs – Computes the magnitude of a complex number. • angle – Computes the phase angle of a complex number. • stem – Draws discrete plots (as opposed to plot, which draws continuous plots). 3 Signal Synthesis Later on in this lab you will use a Fourier series to approximate a square wave. The Fourier series represents the sum of smooth sinusoids, but a square wave is full of sharp edges, so this is a counter-intuitive result! Let’s start with something a little easier: synthesizing the sound of a trumpet. Many musical instruments produce very periodic waveforms. In particular, the pitch of certain notes is determined by specific frequencies, such as 440Hz (which is Concert A, for tuning strings). 1Last revision: Mon May 3 02:48:54 EDT 2010 2http://www.creativecommons.org/licenses/by/2.0 3.1 Trumpet Synthesizer First, download the trumpet sound sample called trumpet.mat from the lab webpage3. This trumpet sound was sampled at a rate of Fs = 11,025 Hz. Exercise 1: • Play the trumpet sound to verify your setup. • Plot a small section of the trumpet sound (100 samples). This will show about three periods of the sampled signal. Is the signal periodic? • Look at the frequency spectrum of the trumpet sound by doing the following: >> Fs = 11025; % the sample rate is 11025 Hz >> Y = fft(trumpet, 512); % take the FFT of trumpet >> Ymag = abs(Y); % take the magnitude of Y >> f = Fs * (0:256)/512; % scale the axis to be meaningful >> plot(f, Ymag(1:257)); % plot Ymag (only half the points are needed) >> xlabel(’Frequency (Hz)’) >> ylabel(’Magnitude’) The series of peaks that you see are the harmonics of the instrument. • Use the “data cursor” tool in MATLAB’s figure window to read the graph data. Write down the frequency and magnitude (the strength of the harmonic) for five to ten of the strongest peaks. You’ll now synthesize the trumpet using only the information from these peaks. Each peak represents a scaled cosine at a particular frequency. The trumpet sound is the sum of these (possibly infinitely many) scaled cosines. Exercise 2: • Create a function called addcosines that will add several cosines together and normalize the output. addcosines should take in three vectors: a time vector time, a vector of frequencies freqs, and a vector of magnitudes mags. The output should have the same dimensions as time. Furthermore, the output should only take on the range of values [-1,1] because you will want to play it like a sound vector. • Verify your code by passing the inputs time = 0:1/Fs:1, freqs = [100 150], and mags = [1 2] to addcosines. Plot the inputs and output. Are they what you expect? 3Or from http://cnx.org/content/m13854/latest/trumpet.mat 2 Here are some hints for the above: • Assume the time vector will have the form 0:1/Fs:duration, where duration is in seconds. • Each cosine function should look like mag(i)*cos(2*pi*freq(i)*t). • Use a for loop to add together the cosines. You’ll have one cosine for each fre- quency/magnitude pair in the freqs and mags vectors. • To scale the output, recall that you did an identical thing in your mixer from Lab 2. Alternatively, you may use soundsc this time. Exercise 3: • Use your addcosines function to sum the cosines specified by the peak data you collected from the trumpet sound in the previous exercise. • Play trumpet and your new synthesized sound. Do they sound the same? • Use subplot to plot a small section of your new synthesized sound along with trumpet. Do they look the same? • Try synthesizing trumpet with fewer frequencies, then try again with more frequencies. • How does this affect the sound of your synthesized trumpet? 3.2 That Funny Phase Although the waveforms of your synthesized trumpet sounds look different from the origi- nal, they ought to sound pretty similar. This should convince you that the Fourier series coefficients are a valid approximation of the original signal! Let’s look at another example: Exercise 4: • Pick two harmonic frequencies and create two cosines, cos1 and cos2, at these frequencies. Use the time vector time = 0:1/Fs:1 with Fs = 8000, but remember that you can only reproduce frequencies that are less than Fs/2. • Add cos1 and cos2 to produce a signal sig1. • Generate cos2b by time-delaying cos2 by half a cycle. • Now create a second signal sig2 that is the sum of cos1 and cos2b. • Use subplot to show a few cycles of sig1 and sig2. How do they compare? What did the time delay do to the phase of sig2? • Play both sig1 and sig2 with soundsc. How do they compare? • Make a couple other variants of sig2 (call them sig2b, sig2c, etc) with different delays on cos2, and compare the outputs to the original sig1. 3 Exercise 5: • Create a third cosine cos3 at an arbitrary frequency. • Add cos3 with a quarter-cycle-delayed version of itself to generate sig3. • Use subplot to show a few cycles of cos3 and sig3. How do they compare? • Play both cos3 and sig3 with soundsc. How do they compare? • What can you conclude about human hearing capabilities? 4 Truncated Fourier Series In this exercise you’ll reconstruct the periodic function x(t), shown below in Figure 1, by synthesizing a periodic signal from a number of Fourier Series coefficients. Along the way, you’ll observe some similarities and differences between your reconstruction and the original signal. Figure 1: A periodic signal. 4.1 Gibbs Phenomena Download the Ck.m function from the lab webpage4. This function takes one argument k , and computes the kth Fourier series coefficient for the square wave in Figure 1. The formula for the coefficient is: ( 0 if k = 0, or k is even; Ck = 1 2kπ kπ jkπ [cos 3 − cos 3 ] if k is odd. 4Or from http://cnx.org/content/m13854/latest/Ck.m 4 −1 For example, Ck(1) = jπ = 0 + j0.3183. Exercise 6: • Compute the coefficients Ck for k ∈ {−10, −9,..., 9, 10}. • Use subplot to show the magnitude and phase (versus frequency ω) for each coefficient. The coefficients are a function of k, which is integer-valued, so use stem to draw the plots. You’ve seen that a signal x(t) can be approximated with a truncated Fourier series. The actual calculation, for a given coefficient range k ∈ {−Kmax, −Kmax +1,...,Kmax −1,Kmax} is as follows: Kmax X jkω0t x(t) = Cke k=−Kmax Kmax X = 2|Ck| cos (kω0t + ∠Ck) k=0 The reason for using the cosine form is to avoid numerical problems and ensure a real answer. For this example, ω0 = 1. Exercise 7: • Implement a function that approximates x(t) with a truncated Fourier series, as described above. Show the TA your (commented) code. • Compute x(t) over the time vector t = -6:.01:6 for each of the following three cases: Kmax = 5, 15, and 30. • Use subplot to plot all three approximations in the same figure. How do the vertical and horizontal edges compare? As you add more cosines to the approximation, you do get closer to the square wave (in terms of squared error). However, at the edges there is some undershoot and overshoot that becomes shorter in time, but the magnitude of the undershoot and overshoot stay large. This persistent undershoot and overshoot at edges is called the Gibbs Phenomenon. In general, this kind of “ringing” occurs at discontinuities – the sharp vertical jumps in the square wave – when you try to synthesize a sharp edge out of too few low frequencies. Another view is that low-pass filtering will trigger the ringing or Gibbs Phenomenon. If you start with a real signal and filter out its higher frequencies, then this is equivalent to going the other way and synthesizing the signal from low frequencies– which is what you just did in the last exercise! As an example, higher frequencies are usually removed when compressing an audio signal (that is, the audio signal is low-pass filtered). Then, if there is an impulse edge or ”attack” in the music, ringing will occur. However, the ringing (called ”pre-echo” in audio) can be 5 heard only before the attack, because the attack masks the ringing that comes after it (this masking effect is psychoacoustic; it happens in your head). High-quality MP3 systems put a lot of effort into detecting attacks and processing the signals to avoid pre-echo. 5 Links This applet5, from the National Taiwan University, provides a great interface for listening to sinusoids and their harmonics. eof 5http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=33 6.
Load more

Recommended publications
  • GIBBS PHENOMENON SUPPRESSION and OPTIMAL WINDOWING for ATTENUATION and Q MEASUREMENTS* Cheh Pan Department of Geophysics, Stanfo
    SLAC-PUB-6222 September 1993 (Ml GIBBS PHENOMENON SUPPRESSION AND OPTIMAL WINDOWING FOR ATTENUATION AND Q MEASUREMENTS* Cheh Pan Department of Geophysics, Stanford University Stanford, CA 94305-2215 and Stanford Linear Accelerator Center Stanford, CA 94309 INTRODUCTION There are basically four known techniques, spectral ratios (Hauge 1981, Johnston and Toksoz 1981, Moos 1984, Goldberg et al 1985, Patten 1988, Sams and Goldberg 1990), forward modeling (Chuen and Toksoz 1981), inversion (Cheng et al 1986), and first pulse rise time (Gladwin and Stacey 1974, Moos 1984), that have been used in the past to measure the attenuation coefficient or quality factor from seismic and acoustic data. Problems have been encountered in using the spectral techniques, which included: (1) the correction for geometrical divergence of the acoustic wave front; (2) the suppression of the , Gibbs phenomenon or the ringing effect in the spectra; and (3) the elimination of contamination from interfering wave modes. Geometrical corrections have been presented by Patten (1988) and Sams and Goldberg (1990) for the borehole acoustics case. The remaining difficulties in the application of the spectral ratios technique come mainly from the suppression of the Gibbs phenomenon and optimal windowing of wave modes. This report deals with these two problems. e - Submitted to Geophysics * Work supP;orted by Department of Energy contract DE-ACO3-76SFOO5 15 : I FUNDAMENTALS The amplitudes R&j) of a seismic signal at frequencyfrecorded by a receiver at a distance or offset x from the source can be represented as, R(x,f)= A(f)G( (1) where A(/) is the source term, G(x) is the geometrical divergence which is assumed to be independent of frequency, and a is the attenuation coefficient.
    [Show full text]
  • Homework 1 Due February 2
    Math 262 / CME 372 Winter 2021 Homework 1 Due February 2 1. Gibbs Phenomenon. Gibbs' phenomenon has to do with how poorly Fourier series converge in the vicinity of a jump or discontinuity of a signal f. This fact was pointed out by Gibbs in a letter to Nature (1899). (Actually Gibb's phenomenon was first described by the British mathematician Wilbraham (1848).) The function Gibbs considered was a sawtooth (1). Gibbs was replying to a letter by the physicist Michaelson to Nature (1898), in which the latter expressed himself doubtful as to the idea that a real discontinuity (in f) can a replace a sum of continuous curves (Sn(f)). Recall that the Fourier series of a periodic function f on the unit interval is Z 1 X i2πkt −i2πkt f(t) = ck e with ck = f(t)e dt: 0 k2Z To investigate Gibb's phenomenon, let us look at the function on the unit interval ( t 0 ≤ t < 1=2 f(t) = (1) t − 1 1=2 ≤ t < 1 (a) Calculate the Fourier coefficients (ck) of f. P i2πkt (b) Let Sn(f) be the partial sum Sn = jk|≤n cke . Calculate the approximation error kf − Sn(f)kL2 as accurately as possible. (c) Consider now −1 0 ≤ t < 1=2 f(t) = : 1 1=2 ≤ t < 1 Gibbs observed that in the vicinity of the jump, the partial sums always overshoot the mark by about 9%. Verify this assertion by carefully setting up a numerical experiment. (d) Repeat the last question on the sawtooth signal (1).
    [Show full text]
  • Fourier Analysis
    FOURIER ANALYSIS Lucas Illing 2008 Contents 1 Fourier Series 2 1.1 General Introduction . 2 1.2 Discontinuous Functions . 5 1.3 Complex Fourier Series . 7 2 Fourier Transform 8 2.1 Definition . 8 2.2 The issue of convention . 11 2.3 Convolution Theorem . 12 2.4 Spectral Leakage . 13 3 Discrete Time 17 3.1 Discrete Time Fourier Transform . 17 3.2 Discrete Fourier Transform (and FFT) . 19 4 Executive Summary 20 1 1. Fourier Series 1 Fourier Series 1.1 General Introduction Consider a function f(τ) that is periodic with period T . f(τ + T ) = f(τ) (1) We may always rescale τ to make the function 2π periodic. To do so, define 2π a new independent variable t = T τ, so that f(t + 2π) = f(t) (2) So let us consider the set of all sufficiently nice functions f(t) of a real variable t that are periodic, with period 2π. Since the function is periodic we only need to consider its behavior on one interval of length 2π, e.g. on the interval (−π; π). The idea is to decompose any such function f(t) into an infinite sum, or series, of simpler functions. Following Joseph Fourier (1768-1830) consider the infinite sum of sine and cosine functions 1 a0 X f(t) = + [a cos(nt) + b sin(nt)] (3) 2 n n n=1 where the constant coefficients an and bn are called the Fourier coefficients of f. The first question one would like to answer is how to find those coefficients.
    [Show full text]
  • ON the GIBBS PHENOMENON and ITS RESOLUTION 1. Introduction. the Gibbs Phenomenon, As We View It, Deals with the Issue of Recover
    SIAM REV. c 1997 Society for Industrial and Applied Mathematics Vol. 39, No. 4, pp. 644–668, December 1997 004 ON THE GIBBS PHENOMENON AND ITS RESOLUTION∗ DAVID GOTTLIEB† AND CHI-WANG SHU† Abstract. The nonuniform convergence of the Fourier series for discontinuous functions, and in particular the oscillatory behavior of the finite sum, was already analyzed by Wilbraham in 1848. This was later named the Gibbs phenomenon. This article is a review of the Gibbs phenomenon from a different perspective. The Gibbs phenomenon, as we view it, deals with the issue of recovering point values of a function from its expansion coefficients. Alternatively it can be viewed as the possibility of the recovery of local infor- mation from global information. The main theme here is not the structure of the Gibbs oscillations but the understanding and resolution of the phenomenon in a general setting. The purpose of this article is to review the Gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function, with the same order of accuracy as in the smooth case. This is done by using the finite expansion series to construct a different, rapidly convergent, approximation. Key words. Gibbs phenomenon, Galerkin, collocation, Fourier, Chebyshev, Legendre, Gegen- bauer, exponential accuracy AMS subject classifications. 42A15, 41A05, 41A25 PII. S0036144596301390 1. Introduction. The Gibbs phenomenon, as we view it, deals with the issue of recovering point values of a function from its expansion coefficients. A prototype problem can be formulated as follows: ˆ Given 2N +1 Fourier coefficients fk, for N k N, of an unknown function f(x) defined everywhere in 1 x 1, construct,− ≤ accurately,≤ point values of the function.
    [Show full text]
  • Higher-Order Wavelet Reconstruction/Differentiation Filters and Gibbs Phenomena
    Journal of Computational Physics 305 (2016) 244–262 Contents lists available at ScienceDirect Journal of Computational Physics www.elsevier.com/locate/jcp Higher-order wavelet reconstruction/differentiation filters and Gibbs phenomena Richard Lombardini 1, Ramiro Acevedo 2, Alexander Kuczala 3, Kerry P. Keys 4, ∗ Carl P. Goodrich 5, Bruce R. Johnson Department of Chemistry, Smalley-Curl Institute and Laboratory for NanoPhotonics, Rice University, Houston, TX 77005, USA a r t i c l e i n f o a b s t r a c t Article history: An orthogonal wavelet basis is characterized by its approximation order, which relates to Received 5 December 2014 the ability of the basis to represent general smooth functions on a given scale. It is known, Received in revised form 18 October 2015 though perhaps not widely known, that there are ways of exceeding the approximation Accepted 23 October 2015 order, i.e., achieving higher-order error in the discretized wavelet transform and its inverse. Available online 30 October 2015 The focus here is on the development of a practical formulation to accomplish this Keywords: first for 1D smooth functions, then for 1D functions with discontinuities and then for Wavelets multidimensional (here 2D) functions with discontinuities. It is shown how to transcend Projection both the wavelet approximation order and the 2D Gibbs phenomenon in representing Reconstruction electromagnetic fields at discontinuous dielectric interfaces that do not simply follow the Gibbs wavelet-basis grid. Multidimensional © 2015 Elsevier Inc. All rights reserved. Boundary 1. Introduction Wavelets are of general interest in developing systematically-improvable multiscale methods for solving differential equa- tions in quantum mechanics, electromagnetism and many other applications.
    [Show full text]
  • The Gibbs-Phenomenon in Option Pricing Methods
    Delft University of Technology Faculty of Electrical Engineering, Mathematics and Computer Science Delft Institute of Applied Mathematics The Gibbs phenomenon in option pricing methods Filtering and other techniques applied to the COS method A thesis submitted to the Delft Institute of Applied Mathematics in partial fulfillment of the requirements for the degree MASTER OF SCIENCE in APPLIED MATHEMATICS by Mark Versteegh Delft, Netherlands October 2012 Copyright c 2012 by Mark Versteegh. All rights reserved MSc thesis APPLIED MATHEMATICS The Gibbs phenomenon in option pricing methods Filtering and other techniques applied to the COS method MARK VERSTEEGH Delft University of Technology Daily Supervisors Responsible professor Prof.dr.ir. C.W. Oosterlee Prof.dr.ir. C.W. Oosterlee M.J. Ruijter, MSc. Other thesis committee members Dr.ir. R.J.Fokkink Dr.ir. J.K. Ryan October, 2012 Delft Table of Contents Preface ix 1 Introduction1 1-1 Option pricing.................................... 1 1-2 Contents of the thesis................................ 4 2 Problem description5 2-1 Introduction to Fourier series and the Fourier transform.............. 5 2-1-1 Fourier series................................ 5 2-1-2 Fourier transform.............................. 7 2-1-3 Characteristic functions........................... 8 2-2 Convergence..................................... 9 2-2-1 Convergence rates.............................. 9 2-2-2 Convergence of the Fourier series...................... 10 2-3 Gibbs phenomenon................................. 11 2-3-1 Introduction................................. 11 2-3-2 Gibbs constant................................ 12 2-3-3 Gibbs phenomenon in applications..................... 13 2-4 COS method..................................... 14 2-4-1 Truncation Range.............................. 18 2-4-2 Error Analysis................................ 18 2-5 Models of interest and Gibb’s phenomenon..................... 19 2-5-1 The Black-Scholes Model.........................
    [Show full text]
  • An Adaptive Approach to Gibbs' Phenomenon
    The University of Southern Mississippi The Aquila Digital Community Master's Theses Summer 2020 An Adaptive Approach to Gibbs’ Phenomenon Jannatul Ferdous Chhoa Follow this and additional works at: https://aquila.usm.edu/masters_theses Part of the Numerical Analysis and Computation Commons, and the Signal Processing Commons Recommended Citation Chhoa, Jannatul Ferdous, "An Adaptive Approach to Gibbs’ Phenomenon" (2020). Master's Theses. 762. https://aquila.usm.edu/masters_theses/762 This Masters Thesis is brought to you for free and open access by The Aquila Digital Community. It has been accepted for inclusion in Master's Theses by an authorized administrator of The Aquila Digital Community. For more information, please contact [email protected]. AN ADAPTIVE APPROACH TO GIBBS’ PHENOMENON by Jannatul Ferdous Chhoa A Dissertation Submitted to the Graduate School, the College of Arts and Sciences and the School of Mathematics and Natural Sciences of The University of Southern Mississippi in Partial Fulfillment of the Requirements for the Degree of Master of Science Approved by: Dr. James Lambers, Committee Chair Dr. Haiyan Tian Dr. Huiqing Zhu Dr. James Lambers Dr. Bernd Schroeder Dr. Karen S. Coats Committee Chair Director of School Dean of the Graduate School August 2020 COPYRIGHT BY JANNATUL FERDOUS CHHOA 2020 ABSTRACT Gibbs’ Phenomenon, an unusual behavior of functions with sharp jumps, is encountered while applying the Fourier Transform on them. The resulting reconstructions have high frequency oscillations near the jumps making the reconstructions far from being accurate. To get rid of the unwanted oscillations, we used the Lanczos sigma factor to adjust the Fourier series and we came across three cases.
    [Show full text]
  • Gibbs Phenomenon and Its Removal for a Class of Orthogonal Expansions
    Gibbs phenomenon and its removal for a class of orthogonal expansions Ben Adcock DAMTP, Centre for Mathematical Sciences University of Cambridge Wilberforce Rd, Cambridge CB3 0WA United Kingdom August 5, 2010 Abstract We detail the Gibbs phenomenon and its resolution for the family of orthogonal ex- pansions consisting of eigenfunctions of univariate polyharmonic operators equipped with homogeneous Dirichlet boundary conditions. As we establish, it is possible to completely describe this phenomenon, including determining exact values for the size of the overshoot near both the domain boundary and the interior discontinuities of the function. Next, we demonstrate how the Gibbs phenomenon can be removed from such expansions using a number of different techniques. As a by-product, we introduce a generalisation of the classical Lidstone polynomials. 1 Introduction Fourier series lie at the heart of countless methods in computational mathematics. Unfortunately, whenever a piecewise smooth function is represented by its Fourier series, the approximation suf- fers from the well-known Gibbs phenomenon [24, 36]. Several characteristics of this phenomenon include the slow convergence of the expansion away from the discontinuity locations, the lack of uniform convergence and the presence of O (1) oscillations near discontinuities [38]. In particu- lar, the maximal overshoot of the Fourier series of a function f near any discontinuity x0 is of + − size c[f(x0 ) − f(x0 )], where c ≈ 0:0895. It is a testament to the importance of the Gibbs phenomenon that the development of techniques for its amelioration, and indeed, complete removal, remains an active area of inquiry. The list of existing methods includes filtering [36], Gegenbauer reconstruction [20, 21], techniques based on extrapolation [14, 15, 16], Pad´emethods [13] and Fourier extension/continuation meth- ods [10, 22], to name but a few (for a more comprehensive survey see [11, 36] and references therein).
    [Show full text]
  • 5: Gibbs Phenomenon Extension: Method (A) T2 Periodic Extension: Method (B) Summary
    5: Gibbs ⊲ Phenomenon Discontinuities Discontinuous Waveform Gibbs Phenomenon Integration Rate at which coefficients decrease with m Differentiation Periodic Extension t2 Periodic 5: Gibbs Phenomenon Extension: Method (a) t2 Periodic Extension: Method (b) Summary E1.10 Fourier Series and Transforms (2014-5559) Gibbs Phenomenon: 5 – 1 / 11 Discontinuities 5: Gibbs Phenomenon A function, v(t), has a discontinuity of amplitude b at t = a if ⊲ Discontinuities Discontinuous Waveform lime→0 (v(a + e) − v(a − e)) = b 6= 0 Gibbs Phenomenon Integration Rate at which Conversely, v(t), is continuous at t = a if the limit, b, equals zero. coefficients decrease with m Differentiation Periodic Extension Examples: t2 Periodic Extension: Method b b (a) u(t) v(t) t2 Periodic 0 0 Extension: Method a – e a a + e a – e a a + e (b) Time (t) Time (t) Summary Continuous Discontinuous We will see that if a periodic function, v(t), is discontinuous, then its Fourier series behaves in a strange way. E1.10 Fourier Series and Transforms (2014-5559) Gibbs Phenomenon: 5 – 2 / 11 Discontinuous Waveform 5: Gibbs Phenomenon Pulse: T = 1 = 20, width= 1 T , height A = 1 1 Discontinuities F 2 0.5 Discontinuous . T 0 ⊲ Waveform 1 0 5 −i2πmFt 0 5 10 15 20 Gibbs Phenomenon Um = Ae dt T 1 0 max(u )=0.500 Integration 0 0.5T 0.5 i −i2πmFt N=0 Rate at which = R e 0 coefficients decrease 2πmFT 0 0 5 10 15 20 with m m i i 1 −iπm ((−1) −1) max(u )=1.137 Differentiation e 1 = 2πm − 1 = 2πm 0.5 N=1 Periodic Extension 0 t2 Periodic 0 5 10 15 20 Extension: Method 0 m 6= 0, even 1 max(u )=1.100 (a) 3 0.5 2 = 0.5 m = 0 N=3 t Periodic 0 Extension: Method −i 0 5 10 15 20 (b) mπ m odd 1 max(u )=1.094 Summary 5 0.5 1 2 1 N=5 So, u(t) = + sin 2πFt + sin 6πFt 0 2 π 3 0 5 10 15 20 1 1 + sin 10πFt + ..
    [Show full text]
  • Gibbs Phenomenon in Engineering Systems
    Gibbs Phenomenon and its Applications in Science and Engineering Josué Njock Libii Engineering Department Indiana University-Purdue University Fort Wayne Fort Wayne, Indiana, 46805-1499 [email protected] Abstract Gibbs phenomenon arises in many applications. In this article, the author first discusses a brief history of this phenomenon and several of its applications in science and engineering. Then, using the Fourier series of a square-wave function and computer software in a classroom exercise, he illustrates how Gibbs phenomenon can be used to illustrate to undergraduate students the concept of nonuniform convergence of successive partial sums over the interval from 0 to B. 1. Introduction Gibbs Phenomenon is intimately related to the study of Fourier series. When a periodic function f(x) with a jump discontinuity is represented using a Fourier series, for example, it is observed that calculating values of that function using a truncated series leads to results that oscillate near the discontinuity [12]. As one includes more and more terms into the series, the oscillations persist but they move closer and closer to the discontinuity itself. Indeed, it is found that the series representation yields an overshoot at the jump, a value that is consistently larger in magnitude than that of the actual function at the jump. No matter how many terms one adds to the series, that overshoot does not disappear. Thus, partial sums that approximate f(x) do not approach f(x) uniformly over an interval that contains a point where the function is discontinuous [23]. This behavior, which appears in many practical applications, is known as Gibbs Phenomenon; it is a common example that is used to illustrate how nonuniform convergence can arise [3].
    [Show full text]
  • Applications of Gibbs Phenomenon in Engneering Education
    Session 3265 Applications of Gibbs Phenomenon in Engineering Education Josué Njock Libii Engineering Department Indiana University-Purdue University Fort Wayne Fort Wayne, Indiana, 46805-1499 1. Introduction. Nonuniform convergence arises in many engineering applications. In this article, the author demonstrates how Gibbs phenomenon associated with the Fourier series of a square-wave function and computer software can be used to illustrate the concept of nonuniform convergence of successive partial sums over the interval from 0 to π. Trigonometric series of the form ¥ a0 f (x) = + å(an cosnx + bn sin nx) 2 n=1 are important in many problems in physics, engineering and biology. Here, a0 ,an ,bn , are coefficients to be determined. Such series and periodic functions that can be represented by them arise naturally in many practical applications. Yamashiro and Grudins[20] used Fourier series to model the flow of air in the lungs; many electric sources of energy generate waveforms that are periodic; sweep generators used to control electron beams in cathode-ray oscilloscopes produce triangular waves; nonlinearities that arise in circuits that are supposed to be linear create periodic functions; and when a nonfiltered electronic rectifier is driven by a sinusoidal source, it produces sine waves that are rectified [21]. Fourier series are used in a wide variety of other fields including electromagnetic wave theory [6], heat transfer [9], acoustics [23], mechanical and structural vibrations [11], and frequency analysis of signals [5]. In contrast to Taylor series, which can only be used to represent functions that have many derivatives, trigonometric series can be used to represent functions that are continuous as well as Page 8.228.1 Page "Proceedings of the 2003 American Society for Engineering Education Annual Conference & Exposition Copyright © 2003, American Society1 for Engineering Education" those that are discontinuous.
    [Show full text]
  • The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations Mathematics and Its Applications
    The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations Mathematics and Its Applications Managing Editor: M.HAZEWINKEL Centre/or Mathematics and Computer Science, Amsterdam, The Netherlands Volume 446 The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations by Abdul J. Jerri Department of Mathematics and Computer Science, Clarkson University, Potsdam, New York, U.S.A. Springer·Science+Business Media, B.V A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-1-4419-4800-7 ISBN 978-1-4757-2847-7 (eBook) DOI 10.1007/978-1-4757-2847-7 Softcover reprint of the hardcover 1st edition 1998 Printed on acid-free paper All Rights Reserved © 1998 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1998 . No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner To my caring wife Suad with gratitude Contents Preface xiii ACKNOWLEDGEMENTS xvii AIM OF THE BOOK xix 1 INTRODUCTION 1 1.1 The Gibbs-Wilbraham Phenomenon 1 1.2 Some Basic Elements of Fourier Analysis. 3 1.3 Illustrations and Analysis . 12 A. The Truncated Fourier Series Approximation 12 B. The Truncated Fourier Integral Approximation 16 1.4 Filtering via the Fejer Averaging 26 A. The Fejer Averaging ..... 28 B. The (C, a) Summability ... 31 1.5 The Lanczos-Local-Type Filtering 34 2 ANALYSIS AND FILTERING 37 2.1 The Truncated Fourier Integral .
    [Show full text]